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Properties

Label 1-388-388.19-r0-0-0
Degree $1$
Conductor $388$
Sign $0.286 - 0.957i$
Analytic cond. $1.80186$
Root an. cond. $1.80186$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.923 − 0.382i)3-s + (0.555 − 0.831i)5-s + (−0.555 − 0.831i)7-s + (0.707 + 0.707i)9-s + (0.923 + 0.382i)11-s + (0.831 + 0.555i)13-s + (−0.831 + 0.555i)15-s + (0.831 + 0.555i)17-s + (0.555 − 0.831i)19-s + (0.195 + 0.980i)21-s + (−0.980 + 0.195i)23-s + (−0.382 − 0.923i)25-s + (−0.382 − 0.923i)27-s + (0.980 − 0.195i)29-s + (−0.382 + 0.923i)31-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)3-s + (0.555 − 0.831i)5-s + (−0.555 − 0.831i)7-s + (0.707 + 0.707i)9-s + (0.923 + 0.382i)11-s + (0.831 + 0.555i)13-s + (−0.831 + 0.555i)15-s + (0.831 + 0.555i)17-s + (0.555 − 0.831i)19-s + (0.195 + 0.980i)21-s + (−0.980 + 0.195i)23-s + (−0.382 − 0.923i)25-s + (−0.382 − 0.923i)27-s + (0.980 − 0.195i)29-s + (−0.382 + 0.923i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(388\)    =    \(2^{2} \cdot 97\)
Sign: $0.286 - 0.957i$
Analytic conductor: \(1.80186\)
Root analytic conductor: \(1.80186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{388} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 388,\ (0:\ ),\ 0.286 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8818904024 - 0.6565399399i\)
\(L(\frac12)\) \(\approx\) \(0.8818904024 - 0.6565399399i\)
\(L(1)\) \(\approx\) \(0.8886117585 - 0.3142982310i\)
\(L(1)\) \(\approx\) \(0.8886117585 - 0.3142982310i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
97 \( 1 \)
good3 \( 1 + (-0.923 - 0.382i)T \)
5 \( 1 + (0.555 - 0.831i)T \)
7 \( 1 + (-0.555 - 0.831i)T \)
11 \( 1 + (0.923 + 0.382i)T \)
13 \( 1 + (0.831 + 0.555i)T \)
17 \( 1 + (0.831 + 0.555i)T \)
19 \( 1 + (0.555 - 0.831i)T \)
23 \( 1 + (-0.980 + 0.195i)T \)
29 \( 1 + (0.980 - 0.195i)T \)
31 \( 1 + (-0.382 + 0.923i)T \)
37 \( 1 + (0.195 - 0.980i)T \)
41 \( 1 + (-0.195 - 0.980i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (-0.923 + 0.382i)T \)
59 \( 1 + (0.980 + 0.195i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.831 - 0.555i)T \)
71 \( 1 + (0.195 - 0.980i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (0.555 - 0.831i)T \)
89 \( 1 + (-0.923 - 0.382i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.846172164164941355221461436206, −23.54484198514115243355188119246, −22.57991219237964131685722146527, −22.28237669085536192278763626634, −21.4446317638075101714288056511, −20.53146903133133595165924262420, −19.11525211875145403293516716008, −18.35472137686404314892499692826, −17.81826893593941036995455124003, −16.5878426113584705828545883935, −16.01377028772968789355956414201, −14.96051964728962243360430211303, −14.10424133866320754372569706151, −12.926761750845280166257223451982, −11.868464107218061020376273011096, −11.27955292938437233623868554282, −9.9895423616559495654220536905, −9.67705113831948801543862131320, −8.24450222421864840803482516588, −6.73959951043431805060724311317, −6.05197340764270140345932255647, −5.45308147051839462274673176886, −3.8346535502925752090892553725, −2.94520764184955995346889554738, −1.29864300910991789530308052670, 0.89833923131603693531426030646, 1.76306932418418606969035455669, 3.75961106814335315101018708657, 4.6819465345523750975093156, 5.86549163970147283357418994385, 6.568959939657354256310004742032, 7.57508070812524820796689877823, 8.93306932294757840857140701455, 9.86977957666023758751580880825, 10.73845454269803776786036601834, 11.91908376217257917277718095698, 12.56238180106568653068767089962, 13.54335439723815808220166236041, 14.16681538316664317698110490366, 16.03002298351652295976581908102, 16.32887826642449367848604272020, 17.41919951192059735238995055754, 17.746776538078014647909779256885, 19.1429223743035290305558069462, 19.85022680658379326710895163681, 20.89383096109775876700065596012, 21.813919194639606750119824553447, 22.61508729154660068233876248711, 23.626331960320470188261728518604, 24.00109428880780578054106289900

Graph of the $Z$-function along the critical line